CAT 2017 Slot 1 — DILR Question 26
Answer the following question based on the information given below.
A new airlines company is planning to start operations in a country. The company has identified ten different cities which they plan to connect through their network to start with. The flight duration between any pair of cities will be less than one hour. To start operations, the company has to decide on a daily schedule.
The underlying principle that they are working on is the following:
Any person staying in any of these 10 cities should be able to make a trip to any other city in the morning and should be able to return by the evening of the same day.
Suppose three of the ten cities are to be developed as hubs. A hub is a city which is connected with every other city by direct flights each way, both in the morning as well as in the evening. The only direct flights which will be scheduled are originating and / or terminating in one of the hubs. Then the minimum number of direct flights that need to be scheduled so that the underlying principle of the airline to serve all the ten cities is met without visiting more than one hub during one trip is:
Answer & solution
- A
54
- B
120
96
- D
60
Let us suppose that City 1, City 2 and City 3 are the hubs and City 4, City 5... upto City 10 are 7 of the other 10 cities.
Now City 1, City 2 and City 3 connect with each other in 4 possible ways (as mentioned in the answer to the previous question).
Now 2 out of 3 cities can be a chosen in 3C2 or 3 ways. So total of no ways City 1, City 2 and City 3 connect with each other is 3 × 4 or 12 ways.
Now City 1 will connect with each of City 4, City 5, City 6 ..... City 10 in 4 possible ways (as explained in the previous questions answer).
So, total number of flights between City 1 and the cities 4 to 10 is 28.
Similarly there will be 28 flights each for City 2 and City 3 that will connect it with the 7 cities. So total minimum number of flights between 2 cities will be 12 + 28 + 28 + 28 = 96.
Hence, option (c).